Since 1990s, two independent research directions emerged. In one direction, the goal was to extend Belitskii's result to the infinite-dimensional setting, while the other direction was concerned with differentiable (at the fixed point 0) linearization in the absence of non-resonance conditions.
Concerning the first direction of research, we mention the work [1]}, which established \(C^1\)  linearization result on Banach spaces under appropriate spectral gap and spectral band conditions for the linear part. This result was further extended to the nonautonomous setting in [2]}. In the second direction, van Strien [3]} claimed the linearization result for \(C^2\)  diffeomorphisms on \(\mathbb {R}^d\)  without any non-resonance conditions. The conclusion was that there exists a conjugacy which is simultaneously differentiable at 0 and Hölder
continuous near 0 (weaker than \(C^1\)  smoothness). However, van Strien's proof was found to be incorrect (see [4]}). In [5]}, the authors gave a correct proof of
van Strien's result by using different methods.
